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Several Complex VariablesMSRI PublicationsVolume 37, 1999

Recent Developments in Seiberg–Witten Theoryand Complex Geometry

CHRISTIAN OKONEK AND ANDREI TELEMAN

We dedicate this paper to our wives Christiane and Roxana

for their invaluable help and support during the past two years.

Abstract. In this article, written at the end of 1996, we survey someof the most important results in Seiberg–Witten Theory which are directlyrelated to Algebraic or Kahlerian Geometry. We begin with an introductionto abelian Seiberg–Witten Theory, with special emphasis on the generalizedSeiberg–Witten invariants, which take also into account 1-homology classesof the base manifold. The more delicate case of manifolds with b+ = 1 isdiscussed in detail; we present our universal wall-crossing formula whichshows that, crossing a wall in the parameter space, produces jumps of theinvariants which are of a purely topological nature.

Next we introduce nonabelian Seiberg–Witten equations associated withvery general compact Lie groups, and we describe in detail some of the prop-erties of the moduli spaces of PU(2)-monopoles. The latter play an impor-tant role in our approach to prove Witten’s conjecture. Then we specializeto the case where the base manifold is a Kahler surface, and we presentthe complex geometric interpretation of the corresponding moduli spaces ofmonopoles. This interpretation is another instance of a Kobayashi–Hitchincorrespondence, which is based on the analysis of various types of vortexequations. Finally we explain our strategy for a proof of Witten’s conjecturein an abstract setting, using the algebraic geometric “coupling principle”and “master spaces” to relate the relevant correlation functions.

Contents

Introduction 3921. Seiberg–Witten Invariants 3942. Nonabelian Seiberg–Witten Theory 4023. Seiberg–Witten Theory and Kahler Geometry 411References 426

The authors were partially supported by AGE (Algebraic Geometry in Europe) contractno. ERBCHRXCT940557 (BBW 93.0187), and by SNF grant no. 21-36111.92.

391

392 CHRISTIAN OKONEK AND ANDREI TELEMAN

Introduction

In October 1994, E. Witten revolutionized the theory of 4-manifolds by in-troducing the now famous Seiberg–Witten invariants [Witten 1994]. These in-variants are defined by counting gauge equivalence classes of solutions of theSeiberg–Witten monopole equations, a system of nonlinear PDE’s which describethe absolute minima of a Yang–Mills–Higgs type functional with an abelian gaugegroup.

Within a few weeks after Witten’s seminal paper became available, severallong-standing conjectures were solved, many new and totally unexpected resultswere found, and much simpler and more conceptional proofs of already estab-lished theorems were given.

Among the most spectacular applications in this early period are the solu-tion of the Thom conjecture [Kronheimer and Mrowka 1994], new results aboutEinstein metrics and Riemannian metrics of positive scalar curvature [LeBrun1995a; 1995b], a proof of a 10

8 bound for intersection forms of Spin manifolds [Fu-ruta 1995], and results about the C∞-classification of algebraic surfaces [Okonekand Teleman 1995a; 1995b; 1997; Friedman and Morgan 1997; Brussee 1996].The latter include Witten’s proof of the C∞-invariance of the canonical class of aminimal surface of general type with b+ 6= 1 up to sign, and a simple proof of theVan de Ven conjecture by the authors. Moreover, combining results in [LeBrun1995b; 1995a] with ideas from [Okonek and Teleman 1995b], P. Lupascu recentlyobtained [1997] the optimal characterization of complex surfaces of Kahler typeadmitting Riemannian metrics of nonnegative scalar curvature.

In two of the earliest papers on the subject, C. Taubes found a deep connectionbetween Seiberg–Witten theory and symplectic geometry in dimension four: Hefirst showed that many aspects of the new theory extend from the case of Kahlersurfaces to the more general symplectic case [Taubes 1994], and then he went onto establish a beautiful relation between Seiberg–Witten invariants and Gromov–Witten invariants of symplectic 4-manifolds [Taubes 1995; 1996].

A report on some papers of this first period can be found in [Donaldson 1996].Since the time this report was written, several new developments have taken

place:The original Seiberg–Witten theory, as introduced in [Witten 1994], has been

refined and extended to the case of manifolds with b+ = 1. The structure ofthe Seiberg–Witten invariants is more complicated in this situation, since theinvariants for manifolds with b+ = 1 depend on a chamber structure. Thegeneral theory, including the complex-geometric interpretation in the case ofKahler surfaces, is now completely understood [Okonek and Teleman 1996b].

At present, three major directions of research have emerged:

– Seiberg–Witten theory and symplectic geometry– Nonabelian Seiberg–Witten theory and complex geometry– Seiberg–Witten–Floer theory and contact structures

SEIBERG–WITTEN THEORY AND COMPLEX GEOMETRY 393

In this article, which had its origin in the notes for several lectures which wegave in Berkeley, Bucharest, Paris, Rome and Zurich during the past two years,we concentrate mainly on the second of these directions.

The reader will probably notice that the nonabelian theory is a subject ofmuch higher complexity than the original (abelian) Seiberg–Witten theory; thedifference is roughly comparable to the difference between Yang–Mills theoryand Hodge theory. This complexity accounts for the length of the article. Inrewriting our notes, we have tried to describe the essential constructions assimply as possible but without oversimplifying, and we have made an effort toexplain the most important ideas and results carefully in a nontechnical way; forproofs and technical details precise references are given.

We hope that this presentation of the material will motivate the reader, and webelieve that our notes can serve as a comprehensive introduction to an interestingnew field of research.

We have divided the article in three chapters. In Chapter 1 we give a concisebut complete exposition of the basics of abelian Seiberg–Witten theory in itsmost general form. This includes the definition of refined invariants for manifoldswith b1 6= 0, the construction of invariants for manifolds with b+ = 1, and theuniversal wall crossing formula in this situation.

Using this formula in connection with vanishing and transversality results, wecalculate the Seiberg–Witten invariant for the simplest nontrivial example, theprojective plane.

In Chapter 2 we introduce nonabelian Seiberg–Witten theories for rather gen-eral structure groups G. After a careful exposition of SpinG-structures and G-monopoles, and a short description of some important properties of their modulispaces, we explain one of the main results of the Habiliationsschrift of the secondauthor [Teleman 1996; 1997]: the fundamental Uhlenbeck type compactificationof the moduli spaces of PU(2)-monopoles.

Chapter 3 deals with complex-geometric aspects of Seiberg–Witten theory:We show that on Kahler surfaces moduli spaces of G-monopoles, for unitarystructure groups G, admit an interpretation as moduli spaces of purely holo-morphic objects. This result is a Kobayashi–Hitchin type correspondence whoseproof depends on a careful analysis of the relevant vortex equations. In theabelian case it identifies the moduli spaces of twisted Seiberg–Witten monopoleswith certain Douady spaces of curves on the surface [Okonek and Teleman 1995a].In the nonabelian case we obtain an identification between moduli spaces ofPU(2)-monopoles and moduli spaces of stable oriented pairs; see [Okonek andTeleman 1996a; Teleman 1997].

The relevant stability concept is new and makes sense on Kahler manifoldsof arbitrary dimensions; it is induced by a natural moment map which is closelyrelated to the projective vortex equation. We clarify the connection betweenthis new equation and the parameter dependent vortex equations which had beenstudied in the literature [Bradlow 1991]. In the final section we construct moduli


spaces of stable oriented pairs on projective varieties of any dimension with GITmethods [Okonek et al. 1999]. Our moduli spaces are projective varieties whichcome with a natural C∗-action, and they play the role of master spaces for stablepairs. We end our article with the description of a very general constructionprinciple which we call “coupling and reduction”. This fundamental principleallows to reduce the calculation of correlation functions associated with vectorbundles to a computation on the space of reductions, which is essentially a modulispace of lower rank objects.

Applied to suitable master spaces on curves, our principle yields a conceptionalnew proof of the Verlinde formulas, and very likely also a proof of the Vafa–Intriligator conjecture. The gauge theoretic version of the same principle canbe used to prove Witten’s conjecture, and more generally, it will probably alsolead to formulas expressing the Donaldson invariants of arbitrary 4-manifolds interms of Seiberg–Witten invariants.

1. Seiberg–Witten Invariants

1.1. The Monopole Equations. Let (X, g) be a closed oriented Riemannian4-manifold. We denote by Λp the bundle of p-forms on X and by Ap := A0(Λp)the corresponding space of sections. Recall that the Riemannian metric g definesa Hodge operator ∗ : Λp −→ Λ4−p with ∗2 = (−1)p. Let Λ2 = Λ2

+ ⊕ Λ2− be the

corresponding eigenspace decomposition.A Spinc-structure on (X, g) is a triple τ = (Σ±, ι, γ) consisting of a pair of

U(2)-vector bundles Σ±, a unitary isomorphism ι : det Σ+ −→ det Σ− and anorientation-preserving linear isometry γ : Λ1 −→ RSU(Σ+,Σ−). Here

RSU(Σ+,Σ−) ⊂ HomC(Σ+,Σ−)

is the subbundle of real multiples of (fibrewise) isometries of determinant 1. Thespinor bundles Σ± of τ are — up to isomorphism — uniquely determined by theirfirst Chern class c := c1(det Σ±), the Chern class of the Spinc(4)-structure τ .This class can be any integral lift of the second Stiefel–Whitney class w2(X) ofX, and, given c, we have

c2(Σ±) =14

(c2 − 3σ(X) ∓ 2e(X)).

Here σ(X) and e(X) denote the signature and the Euler characteristic of X.The map γ is called the Clifford map of the Spinc-structure τ . We denote by

Σ the total spinor bundle Σ := Σ+⊕Σ−, and we use the same symbol γ also forthe induced the map Λ1 −→ su(Σ) given by

u 7−→(

0 −γ(u)∗

γ(u) 0

).


Note that the Clifford identity

γ(u)γ(v) + γ(v)γ(u) = −2g(u, v)

holds, and that the formula

Γ(u ∧ v) := 12 [γ(u), γ(v)]

defines an embedding Γ : Λ2 −→ su(Σ) which maps Λ2± isometrically onto

su(Σ±) ⊂ su(Σ).The second cohomology group H2(X,Z) acts on the set of equivalence classes

c of Spinc(4)-structures on (X, g) in a natural way: Given a representative τ =(Σ±, ι, γ) of c and a Hermitian line bundle M representing a class m ∈ H2(X,Z),the tensor product (Σ±⊗M, ι⊗ idM⊗2 , γ⊗ idM ) defines a Spinc-structure τm.Endowed with the H2(X,Z)-action given by (m, [τ ]) 7−→ [τm], the set of equiva-lence classes of Spinc-structures on (X, g) becomes a H2(X,Z)-torsor, which isindependent of the metric g up to canonical isomorphism [Okonek and Teleman1996b]. We denote this H2(X,Z)-torsor by Spinc(X).

Recall that the choice of a Spinc(4)-structure (Σ±, ι, γ) defines an isomorphismbetween the affine space A(det Σ+) of unitary connections in det Σ+ and theaffine space of connections in Σ± which lift the Levi-Civita connection in thebundle Λ2

± ' su(Σ±). We denote by a ∈ A(Σ) the connection corresponding toa ∈ A(det Σ+).

The Dirac operator associated with the connection a ∈ A(det Σ+) is the com-position

6Da : A0(Σ±) ∇a−−→ A1(Σ±) γ−→ A0(Σ∓)

of the covariant derivative ∇a in the bundles Σ± and the Clifford multiplicationγ : Λ1 ⊗Σ± −→ Σ∓.

Note that, in order to define the Dirac operator, one needs a Clifford map,not only a Riemannian metric and a pair of spinor bundles; this will later be-come important in connection with transversality arguments. The Dirac oper-ator 6Da : A0(Σ±) −→ A0(Σ∓) is an elliptic first order operator with symbolγ : Λ1 −→ RSU(Σ±,Σ∓). The direct sum-operator 6Da : A0(Σ) −→ A0(Σ) onthe total spinor bundle is selfadjoint and its square has the same symbol as therough Laplacian ∇∗a∇a on A0(Σ).

The corresponding Weitzenbock formula is

6D2a = ∇∗a∇a + 1

2Γ(Fa) + 1

4s idΣ ,

where Fa ∈ iA2 is the curvature of the connection a, and s denotes the scalarcurvature of (X, g) [Lawson and Michelsohn 1989].

To write down the Seiberg–Witten equations, we need the following nota-tions: For a connection a ∈ A(det Σ+) we let F±a ∈ iA2

± be the (anti) self-dual components of its curvature. Given a spinor Ψ ∈ A0(Σ+), we denote by(ΨΨ)0 ∈ A0(End0(Σ±)) the trace free part of the Hermitian endomorphism


Ψ⊗Ψ. Now fix a Spinc(4)-structure τ = (Σ±, ι, γ) for (X, g) and a closed 2-formβ ∈ A2. The β-twisted monopole equations for a pair (a,Ψ) ∈ A(det Σ+) ×A0(Σ+) are

6DaΨ = 0,

Γ(F+a + 2πiβ+) = (ΨΨ)0 .

(SWτβ)

These β-twisted Seiberg–Witten equations should not be regarded as perturba-tions of the equations (SWτ

0) since later the cohomology class of β will be fixed.The twisted equations arise naturally in connection with nonabelian monopoles(see Section 2.2). Using the Weitzenbock formula one easily gets the followingfact:

Lemma 1.1.1. Let β be a closed 2-form and (a,Ψ) ∈ A(det Σ+)×A0(Σ+). Then

‖6DaΨ‖2 + 14‖(F

+a + 2πiβ+)− (ΨΨ)0‖2

= ‖∇AaΨ‖2 + 14‖F+

a + 2πiβ+‖2 + 18‖Ψ‖4L4 +

∫X

((14s idΣ+ −Γ(πiβ+))Ψ,Ψ).

Corollary 1.1.2 [Witten 1994]. On manifolds (X, g) with nonnegative scalarcurvature s the only solutions of (SWτ

0) are pairs (a, 0) with F+a = 0.

1.2. Seiberg–Witten Invariants for 4-Manifolds with b+ > 1. Let (X, g)be a closed oriented Riemannian 4-manifold, and let c ∈ Spinc(X) be an equiv-alence class of Spinc-structures of Chern class c, represented by the triple τ =(Σ±, ι, γ). The configuration space for Seiberg–Witten theory is the productA(det Σ+)× A0(Σ+) on which the gauge group G := C∞(X, S1) acts by

f ·(a,Ψ) := (a− 2f−1df , fΨ).

Let B(c) :=(A(det Σ+) × A0(Σ+)

)/G be the orbit space; up to homotopy

equivalence, it depends only on the Chern class c. Since the gauge group actsfreely in all points (a,Ψ) with Ψ 6= 0, the open subspace

B(c)∗ :=(A(det Σ+) ×

(A0(Σ+) \ 0

))/G

is a classifying space for G. It has the weak homotopy type of a productK(Z, 2) × K(H1(X,Z), 1) of Eilenberg–Mac Lane spaces and there is a natu-ral isomorphism

ν : Z[u]⊗ Λ∗(H1(X,Z)/

Tors) −→ H∗(B(c)∗,Z),

where the generator u is of degree 2. The G-action on A(det Σ+)×A0(Σ+) leavesthe subset [A(det Σ+) × A0(Σ+)]SWτ

β of solutions of (SWτβ) invariant; the orbit

spaceWτβ := [A(det Σ+)×A0(Σ+)]SWτ

β/G

is the moduli space of β-twisted monopoles. It depends, up to canonical isomor-phism, only on the metric g, on the closed 2-form β, and on the class c ∈ Spinc(X)[Okonek and Teleman 1996b].


Let Wτβ∗ ⊂Wτ

β be the open subspace of monopoles with nonvanishing spinor-component; it can be described as the zero-locus of a section in a vector-bundleover B(c)∗. The total space of this bundle is[

A(det Σ+)× (A0(Σ+) \ 0)]×G

[iA2

+ ⊕A0(Σ−)],

and the section is induced by the G-equivariant map

SWτβ : A(det Σ+)×

(A0(Σ+) \ 0

)−→ iA2

+ ⊕ A0(Σ−)

given by the equations (SWτβ).

Completing the configuration space and the gauge group with respect to suit-able Sobolev norms, we can identify Wτ

β∗ with the zero set of a real analytic

Fredholm section in the corresponding Hilbert vector bundle on the Sobolevcompletion of B(c)∗, hence we can endow this moduli space with the structureof a finite dimensional real analytic space. As in the instanton case, one has aKuranishi description for local models of the moduli space around a given point[a,Ψ] ∈Wτ

β in terms of the first two cohomology groups of the elliptic complex

0 −→ iA0D0p−−→ iA1 ⊕ A0(Σ+)

D1p−−→ iA2

+ ⊕ A0(Σ−) −→ 0 (Cp)

obtained by linearizing in p = (a,Ψ) the action of the gauge group and theequivariant map SWτ

β. The differentials of this complex are

D0p(f) = (−2df, fΨ),

D1p(α, ψ) =

(d+α− Γ−1[(Ψψ)0 + (ψΨ)0], 6Da(ψ) + γ(α)(Ψ)

),

and its index wc depends only on the Chern class c of the Spinc-structure τ andon the characteristic classes of the base manifold X:

wc = 14 (c2 − 3σ(X) − 2e(X)).

The moduli space Wτβ is compact. This follows, as in [Kronheimer and Mrowka

1994], from the following consequence of the Weitzenbock formula and the max-imum principle.

Proposition 1.2.1 (A priori C0-bound of the spinor component). If

(a,Ψ) is a solution of (SWτβ), then

sup |Ψ|2 ≤ max(0, sup

X(−s+ |4πβ+|)

).

Moreover, let Wτ be the moduli space of triples

(a,Ψ, β) ∈ A(det Σ+)× A0(Σ+)× Z2DR(X)

solving the Seiberg–Witten equations above now regarded as equations for thetriple (a,Ψ, β). Two such triples define the same point in Wτ if they are congru-ent modulo the gauge group G acting trivially on the third component. Usingthe proposition above and arguments of [Kronheimer and Mrowka 1994], one can


easily see that the natural projection Wτ p−→ Z2DR(X) is proper. Moreover, one

has the following transversality results:

Lemma 1.2.2. After suitable Sobolev completions the following results hold :

1 [Kronheimer and Mrowka 1994]. The open subspace [Wτ ]∗ ⊂ Wτ of pointswith nonvanishing spinor component is smooth.

2 [Okonek and Teleman 1996b]. For any de Rham cohomology class b ∈ H2DR(X)

the moduli space [Wτb ]∗ := [Wτ ]∗ ∩ p−1(b) is also smooth.

Now let c ∈ H2(X,Z) be a characteristic element, that is, an integral lift ofw2(X). A pair (g, b) ∈ Met(X) × H2

DR(X) consisting of a Riemannian met-ric g on X and a de Rham cohomology class b is called c-good when the g-harmonic representant of c − b is not g-antiselfdual. This condition guaranteesthat Wτ

β = Wτβ∗ for every Spinc-structure τ of Chern class c and every 2-form β

in b. Indeed, if (a, 0) would solve (SWτβ), then the g-antiselfdual 2-form i

2πFa−βwould be the g-harmonic representant of c − b.

In particular, using the transversality results above, one gets:

Theorem 1.2.3 [Okonek and Teleman 1996b]. Let c ∈ H2(X,Z) be a charac-teristic element and suppose (g, b) ∈ Met(X) × H2

DR(X) is c-good . Let τ be aSpinc-structure of Chern class c on (X, g), and β ∈ b a general representant ofthe cohomology class b. Then the moduli space Wτ

β = Wτβ∗ is a closed manifold

of dimension wc = 14(c2 − 3σ(X) − 2e(X)).

Fix a maximal subspace H2+(X,R) of H2(X,R) on which the intersection form

is positive definite. The dimension b+(X) of such a subspace is the numberof positive eigenvalues of the intersection form. The moduli space Wτ

β can beoriented by the choice of an orientation of the line detH1(X,R)⊗detH2

+(X,R)∨.Let [Wτ

β]O ∈ Hwc(B(c)∗,Z) be the fundamental class associated with thechoice of an orientation O of the line detH1(X,R) ⊗ detH2

+(X)∨.The Seiberg-Witten form associated with the data (g, b, c,O) is the element

SW(g,b)X,O (c) ∈ Λ∗H1(X,Z) defined by

SW(g,b)X,O (c)(l1 ∧ · · · ∧ lr) :=

⟨ν(l1) ∪ · · · ∪ ν(lr) ∪ u(wc−r)/2, [Wτ

β]O

⟩for decomposable elements l1 ∧ · · · ∧ lr with r ≡ wc (mod 2). Here τ is a Spinc-structure on (X, g) representing the class c ∈ Spinc(X), and β is a general formin the class b.

One shows, using again transversality arguments, that the Seiberg–Wittenform SW(g,b)

X,O (c) is well defined, independent of the choices of τ and β. Moreover,if any two c-good pairs (g0, b0), (g1, b1) can be joined by a smooth path of c-goodpairs, then SW(g,b)

X,O (c) is also independent of (g, b) [Okonek and Teleman 1996b].Note that the condition “(g, b) is not c-good” is of codimension b+(X) for a

fixed class c. This means that for manifolds with b+(X) > 1 we have a well


defined map

SWX,O : Spinc(X) −→ Λ∗H1(X,Z)

which associates to a class of Spinc-structures c the form SW(g,b)X,O (c) for any

b ∈ H2DR(X) such that (g, b) is c-good. This map, which is functorial with respect

to orientation preserving diffeomorphisms, is the Seiberg–Witten invariant.Using the identity in Lemma 1.1.1, one can easily prove the next result:

Remark 1.2.4 [Witten 1994]. Let X be an oriented closed 4-manifold withb+(X) > 1. Then the set of classes c ∈ Spinc(X) with nontrivial Seiberg–Witteninvariant is finite.

In the special case b+(X) > 1, b1(X) = 0, SWX,O is simply a function

SWX,O : Spinc(X) −→ Z .

The values SWX,O(c) ∈ Z are refinements of the numbers nOc defined by

Witten [1994]. More precisely:

nO

c =∑

c

SWX,O(c),

the summation being over all classes of Spinc-structures c of Chern class c. It iseasy to see that the indexing set is a torsor for the subgroup Tors2H

2(X,Z) of2-torsion classes in H2(X,Z).

The structure of the Seiberg–Witten invariants for manifolds with b+(X) = 1is more complicated and will be described in the next section.

1.3. The Case b+ = 1 and the Wall Crossing Formula. Let X be aclosed oriented differentiable 4-manifold with b+(X) = 1. In this situation theSeiberg–Witten forms depend on a chamber structure: Recall first that in thecase b+(X) = 1 there is a natural map Met(X) −→ P(H2

DR(X)) which sendsa metric g to the line R[ω+] ⊂ H2

DR(X), where ω+ is any nontrivial g-selfdualharmonic form. Let

H := h ∈ H2DR(X) : h2 = 1

be the hyperbolic space. This space has two connected components, and thechoice of one of them orients the lines H2

+,g(X) of selfdual g-harmonic forms,for all metrics g. Furthermore, once we fix a component H0 of H, every metricdefines a unique g-selfdual form ωg of length 1 with [ωg] ∈H0; see Figure 1.

Let c ∈ H2(X,Z) be characteristic. The wall associated with c is the hyper-surface

c⊥ := (h, b) ∈H×H2DR(X) : (c − b)·h = 0,

and the connected components of [H×H2DR(X)]\c⊥ are called chambers of type

c.


H0

R[ωg]

[ωg]

Figure 1.

Notice that the walls are nonlinear. Each characteristic element c definesprecisely four chambers of type c, namely

CH0,± := (h, b) ∈H×H2DR(X) : ±(c − b)·h < 0, H0 ∈ π0(H),

and each of these four chambers contains elements of the form ([ωg], b) withg ∈Met(X).

Let O1 be an orientation of H1(X,R). The choice of O1 together with thechoice of a component H0 ∈ π0(H) defines an orientation O = (O1,H0) ofdet(H1(X,R)) ⊗ det(H2

+(X,R)∨). Set

SW±X,(O1,H0)(c) := SW(g,b)X,O (c),

where (g, b) is a pair such that ([ωg], b) belongs to the chamber CH0,±. The map

SWX,(O1,H0) : Spinc(X) −→ Λ∗H1(X,Z)× Λ∗H1(X,Z)

which associates to a class c of Spinc-structures on the oriented manifold X thepair of forms (SW+

X,(O1,H0)(c), SW−X,(O1,H0)(c)) is the Seiberg-Witten invariantof X with respect to the orientation data (O1,H0). This invariant is functorial


with respect to orientation-preserving diffeomorphisms and behaves as followswith respect to changes of the orientation data:

SWX,(−O1,H0)(c) = −SWX,(O1,H0)(c), SW±X,(O1,−H0)(c) = −SW∓X,(O1,H0)(c).

More important, however, is the fact that the difference

SW+X,(O1,H0)(c) − SW−X,(O1,H0)(c)

is a topological invariant of the pair (X, c). To be precise, consider the elementuc ∈ Λ2

(H1(X,Z)

/Tors

)defined by

uc(a ∧ b) := 12

⟨a ∪ b ∪ c, [X]

⟩for elements a, b ∈ H1(X,Z). The following universal wall-crossing formulageneralizes results of [Witten 1994; Kronheimer and Mrowka 1994; Li and Liu1995].

Theorem 1.3.1 (Wall-crossing formula [Okonek and Teleman 1996b]). LetlO1 ∈ Λb1H1(X,Z) be the generator defined by the orientation O1, and let r ≥ 0with r ≡ wc (mod 2). For every λ ∈ Λr

(H1(X,Z)

/Tors

)we have[

SW+X,(O1,H0)(c) − SW−X,(O1,H0)(c)

](λ) =

(−1)(b1−r)/2(12(b1 − r)/2

)!〈λ ∧ u(b1−r)/2

c , lO1〉

when r ≤ min(b1, wc), and the difference vanishes otherwise.

We illustrate these results with the simplest possible example, the projectiveplane.

Example. Let P2 be the complex projective plane, oriented as a complex mani-fold, and denote by h the first Chern class of OP2(1). Since h2 = 1, the hyperbolicspace H consists of two points H = ±h. We choose the component H0 := hto define orientations.

An element c ∈ H2(P2,Z) is characteristic if and only if c ≡ h (mod 2). InFigure 2 we have drawn (as vertical intervals) the two chambers

CH0,± = (h, b) ∈H0 ×H2DR(P2) : ±(c− b)·h < 0

of type c, for every c ≡ h (mod 2).The set Spinc(P2) can be identified with the set (2Z + 1)h of characteristic

elements under the map which sends a Spinc-structure c to its Chern class c.The corresponding virtual dimension is wc = 1

4(c2 − 9). Note that, for any

metric g, the pair (g, 0) is c-good for all characteristic elements c. Also recallthat the Fubini–Study metric g is a metric of positive scalar curvature whichcan be normalized such that [ωg] = h. We can now completely determine theSeiberg–Witten invariant SWP2,H0

using three simple arguments:

(i) For c = ±h we have wc < 0, hence SW±P2,H0(c) = 0, by the transversality

results of Section 1.2.


b

c

Figure 2.

(ii) Let c be a characteristic element with wc ≥ 0. Since the Fubini–Study metricg has positive scalar curvature and (g, 0) is c-good, we have Wτ

0 = Wτ0∗ = ∅

by Corollary 1.1.2. But this moduli space can be used to compute SW±P2,H0(c)

for characteristic elements c with ±c·h < 0. Thus we find SW±P2,H0(c) = 0

when wc ≥ 0 and ±c·h < 0.(iii) The remaining values, SW∓P2,H0

(c) = 0 for classes satisfying wc ≥ 0 and±c·h < 0, are determined by the wall-crossing formula. Altogether we get

SW+P2,H0

(c) =

1 if c·h ≥ 3,0 if c·h < 3,

SW−P2,H0(c) =

−1 if c·h ≤ −3,

0 if c·h > −3.

2. Nonabelian Seiberg–Witten Theory

2.1. G-Monopoles. Let V be a Hermitian vector space, and let U(V ) be itsgroup of unitary automorphisms. For any closed subgroup G ⊂ U(V ) whichcontains the central involution − idV , we define a new Lie group by

SpinG(n) := Spin(n) ×Z2 G.

By construction one has the exact sequences

1 −→ Spin −→ SpinG δ−→ G/Z2 −→ 1,

1 −→ G −→ SpinG π−→ SO −→ 1,

1 −→ Z2 −→ SpinG(π,δ)−−−→ SO×G

/Z2 −→ 1,

where Spin, SpinG, SO denote one of the groups Spin(n), SpinG(n), SO(n),respectively.


Given a SpinG-principal bundle PG over a topological space, we form thefollowing associated bundles:

δ(PG) := PG ×δ (G/Z2), G(PG) := PG ×Ad G, |g(PG) := PG ×ad g,

where g stands for the Lie algebra of G. The group G of sections of the bun-dle G(PG) can be identified with the group of automorphism of PG over theassociated SO-bundle PG ×π SO.

Consider now an oriented manifold (X, g), and let Pg be the SO-bundle oforiented g-orthonormal coframes. A SpinG-structure in Pg is a principal bundlemorphism σ : PG −→ Pg of type π [Kobayashi and Nomizu 1963]. An isomor-phism of SpinG-structures σ, σ′ in Pg is a bundle isomorphism f : PG −→ P ′G

with σ′ f = σ. One shows that the data of a SpinG-structure in (X, g) is equiva-lent to the data of a linear, orientation-preserving isometry γ : Λ1 −→ PG×πRn,which we call the Clifford map of the SpinG-structure [Teleman 1997].

In dimension 4, the spinor group Spin(4) splits as

Spin(4) = SU(2)+ × SU(2)− = Sp(1)+ × Sp(1)− .

Using the projectionsp± : Spin(4) −→ SU(2)±

one defines the adjoint bundles

ad±(PG) := PG ×ad± su(2).

Coupling p± with the natural representation ofG in V , we obtain representationsλ± : SpinG(4) −→ U(H± ⊗C V ) and associated spinor bundles

Σ±(PG) := PG ×λ± (H± ⊗C V ).

The Clifford map γ : Λ1 −→ PG ×π R4 of the SpinG-structure yields identifi-cations

Γ : Λ2± −→ ad±(PG).

An interesting special case occurs when V is a Hermitian vector space over thequaternions and G is a subgroup of Sp(V ) ⊂ U(V ). Then one can define realspinor bundles

Σ±R (PG) := PG ×ρ± (H± ⊗H V ),

associated with the representations

ρ± : SpinG(4) −→ SO(H± ⊗H V ).

Examples. Let (X, g) be a closed oriented Riemannian 4-manifold with coframebundle Pg.

G = S1: A SpinS1-structure is just a Spinc-structure as described in Chapter 1

[Teleman 1997].


G = Sp(1): SpinSp(1)-structures have been introduced in [Okonek and Teleman1996a], where they were called Spinh-structures. The map which associatesto a SpinSp(1)-structure σ : P h −→ Pg the first Pontrjagin class p1(δ(P h))of the associated SO(3)-bundle δ(P h), induces a bijection between the set ofisomorphism classes of SpinSp(1)-structures in (X, g) and the set

p ∈ H4(X,Z) : p ≡ w2(X)2 (mod 4).

There is a 1-to-1 correspondence between isomorphism classes of SpinSp(1)-structures in (X, g) and equivalence classes of triples (τ : P S

1 −→ Pg, E, ι)consisting of a SpinS

1-structure τ , a unitary vector bundle E of rank 2, and

an unitary isomorphism ι : det Σ+τ −→ detE. The equivalence relation is

generated by tensorizing with Hermitian line bundles [Okonek and Teleman1996a; Teleman 1997]. The associated bundles are — in terms of these data —given by

δ(P h) = PE/S1 , G(P h) = SU(E), |g(P h) = su(E),

Σ±(P h) = (Σ±τ )∨ ⊗ E, Σ±R (P h) = RSU(Σ±τ , E),

where PE denotes the principal U(2)-frame bundle of E.

G = U(2): In this case G/Z2 splits as G

/Z2 = PU(2) × S1, and we write

δ in the form (δ, det). The map which associates to a SpinU(2)-structureσ : P u −→ Pg the characteristic classes p1(δ(P u)), c1(detP u) identifies theset of isomorphism classes of SpinU(2)-structures in (X, g) with the set

(p, c) ∈ H4(X,Z)×H2(X,Z) : p ≡ (w2(X) + c)2 (mod 4).

There is a one-to-one correspondence between isomorphism classes of SpinU(2)-structures in (X, g) and equivalence classes of pairs (τ : P S

1 −→ Pg, E) con-sisting of a SpinS

1-structure τ and a unitary vector bundle of rank 2. Again

the equivalence relation is given by tensorizing with Hermitian line bundles[Teleman 1997]. If σ : P u −→ Pg corresponds to the pair (τ : P S

1 −→ Pg, E),the associated bundles are now

δ(P u) = PE/S1 , detP u = det[Σ+

τ ]∨ ⊗ detE,

G(P u) = U(E), |g(P u) = u(E), Σ±(P u) = (Σ±τ )∨ ⊗ E.

We will later also need the subbundles G0(P u) := P u×Ad SU(2) ' SU(E) and|g0 := P u×ad su(2) ' su(E). The group of sections Γ(X,G0) can be identifiedwith the group of automorphisms of P u over Pg ×X det(P u).

Now consider again a general SpinG-structure σ : PG −→ Pg in the 4-manifold(X, g). The spinor bundle Σ±(PG) has H± ⊗C V as standard fiber, so that thestandard fiber su(2)± ⊗ g of the bundle ad±(PG)⊗ |g(PG) can be viewed as realsubspace of End(H± ⊗C V ). We define a quadratic map

µ0G : H± ⊗C V −→ su(2)± ⊗ g


by sending ψ ∈ H± ⊗C V to the orthogonal projection prsu(2)±⊗g(ψ ⊗ ψ) of theHermitian endomorphism (ψ ⊗ ψ) ∈ End(H± ⊗C V ). One can show that −µ0G

is the total (hyperkahler) moment map for the G-action on the space H± ⊗C Vendowed with the natural hyperkahler structure given by left multiplication withquaternionic units [Teleman 1997].

These maps give rise to quadratic bundle maps

µ0G : Σ±(PG) −→ ad±(PG) ⊗ |g(PG).

In the case G = U(2) one can project µ0U(2) on ad±(PG)⊗ |g0(PG) and getsa map

µ00 : Σ±(P u) −→ ad±(P u) ⊗ |g0(P u).

Note that a fixed SpinG-structure σ : PG −→ Pg defines a bijection be-tween connections A ∈ A(δ(PG)) in δ(PG) and connections A ∈ A(PG) in theSpinG-bundle PG which lift the Levi-Civita connection in Pg via σ. This followsimmediately from the third exact sequence above. Let

6DA : A0(Σ±(PG)) −→ A0(Σ∓(PG))

be the associated Dirac operator, defined by

6DA : A0(Σ±(PG))∇A−−→ A1(Σ±(PG))

γ−→ A0(Σ∓(PG)).

Here γ : Λ1 ⊗ Σ±(PG) −→ Σ∓(PG) is the Clifford multiplication correspondingto the embeddings γ : Λ1 −→ PG ×π R4 ⊂ HomC(Σ±(PG),Σ∓(PG)).

Definition 2.1.1. Let σ : PG −→ Pg be a SpinG-structure in the Rie-mannian manifold (X, g). The G-monopole equations for a pair (A,Ψ), withA ∈ A(δ(PG)) and Ψ ∈ A0(Σ+(PG)), are

6DAΨ = 0,

Γ(F+A ) = µ0G(Ψ).

(SWσ)

The solutions of these equations will be called G-monopoles. The symmetrygroup of the G-monopole equations is the gauge group G := Γ(X,G(PG)). Ifthe Lie algebra of G has a nontrivial center z(g), then one has a family ofG-equivariant “twisted” G-monopole equations (SWσ

β) parameterized by iz(g)-valued 2-forms β ∈ A2(iz(g)):

6DAΨ = 0,

Γ((FA + 2πiβ)+) = µ0G(Ψ).(SWσ

β)

We denote by Mσ and Mσβ, respectively, the corresponding moduli spaces of

solutions modulo the gauge group G.Since in the case G = U(2) there exists the splitting

U(2)/Z2 = PU(2)× S1 ,


the data of a connection in δ(P u) = δ(P u) ×X detP u is equivalent to the dataof a pair of connections (A, a) ∈ A(δ(P u)) × A(detP u). This can be used tointroduce new important equations, obtained by fixing the abelian connectiona ∈ A(detP u) in the U(2)-monopole equations, and regarding it as a parameter.One gets in this way the equations

6DA,aΨ = 0,

Γ(F+A ) = µ00(Ψ)

(SWσa)

for a pair

(A,Ψ) ∈ A(δ(P u)) ×A0(Σ+(P u)),

which will be called the PU(2)-monopole equations. These equations should beregarded as a twisted version of the quaternionic monopole equations introducedin [Okonek and Teleman 1996a], which coincide in our present framework withthe SU(2)-monopole equations. Indeed, a SpinU(2)-structure σ : P u −→ Pg withtrivialized determinant line bundle can be regarded as SpinSU(2)-structure, andthe corresponding quaternionic monopole equations are (SWσ

θ ), where θ is thetrivial connection in detP u.

The PU(2)-monopole equations are only invariant under the group G0 :=Γ(X,G0) of automorphisms of P u over Pg ×X detP u. We denote by Mσ

a themoduli space of PU(2)-monopoles modulo this gauge group. Note that Mσ

a comeswith a natural S1-action given by the formula ζ ·[A,Ψ] := [A, ζ

12 Ψ].

Comparing with other formalisms:

1. For G = S1, V = C one recovers the original abelian Seiberg–Witten equa-tions and the twisted abelian Seiberg–Witten equations of [LeBrun 1995b;Brussee 1996; Okonek and Teleman 1996b].

2. For G = S1, V = C⊕k one gets the so called “multimonopole equations”studied by J. Bryan and R. Wentworth [1996].

3. In the case G = U(2), V = C2 one obtains the U(2)-monopole equationswhich were studied in [Okonek and Teleman 1995a] (see also Chapter 3).

4. In the case of a Spin-manifoldX and G = SU(2) the corresponding monopoleequations were introduced in [Okonek and Teleman 1995c]; they have beenstudied from a physical point of view in [Labastida and Marino 1995].

5. If X is simply connected, the S1-quotient Mσa

/S1 of a moduli space of PU(2)-

monopoles can be identified with a moduli space of “nonabelian monopoles” asdefined in [Pidstrigach and Tyurin 1995]. Note that in the general non-simplyconnected case, one has to use our formalism.

Remark 2.1.2. Let G = Sp(n)·S1 ⊂ U(C2n) be the Lie group of transforma-tions of H⊕n generated by left multiplication with quaternionic matrices in Sp(n)and by right multiplication with complex numbers of modulus 1. Then G

/Z2

splits as PSp(n) × S1. In the same way as in the PU(2)-case one defines the


PSp(n)-monopole equations (SWσa ) associated with a SpinSp(n)·S1

(4)-structureσ : PG −→ Pg in (X, g) and an abelian connection a in the associated S1-bundle.

The solutions of the (twisted) G- and PU(2)-monopole equations are the absoluteminima of certain gauge invariant functionals on the corresponding configurationspaces A(δ(PG))× A0(Σ+(PG)) and A(δ(P u))× A0(Σ+(PG)).

For simplicity we describe here only the case of nontwisted G-monopoles. TheSeiberg-Witten functional SWσ : A(δ(PG)) × A0(Σ+(PG)) −→ R associated toa SpinG-structure is defined by

SWσ(A,Ψ) := ‖∇AΨ‖2 + 14‖FA‖2 + 1

2‖µ0G(Ψ)‖2 + 1

4

∫X

s|Ψ|2 .

The Euler-Lagrange equations describing general critical points are

d∗AFA + J(A,Ψ) = 0,

∆AΨ + µ0G(Ψ)(Ψ) + 14sΨ = 0,

where the current J(A,Ψ) ∈ A1(|g(PG)) is given by√

32 times the orthogonalprojection of the End(Σ+(PG))-valued 1-form ∇AΨ ⊗ Ψ ∈ A1(End(Σ+(PG)))onto A1(|g(PG)).

In the abelian case G = S1, V = C, a closely related functional and thecorresponding Euler–Lagrange equations have been investigated in [Jost et al.1996].

2.2. Moduli Spaces of PU(2)-Monopoles. We retain the notations of theprevious section. Let σ : P u −→ Pg be a SpinU(2)-structure in a closed orientedRiemannian 4-manifold (X, g), and let a ∈ A(detP u) be a fixed connection. ThePU(2)-monopole equations

6DA,aΨ = 0,

Γ(F+A ) = µ00(Ψ)

(SWσa)

associated with these data are invariant under the action of the gauge groupG0, and hence give rise to a closed subspace Mσ

a ⊂ B(P u) of the orbit spaceB(P u) :=

(A(δ(P u)) ×A0(Σ+(P u))

)/G0.

The moduli space Mσa can be endowed with the structure of a ringed space

with local models constructed by the well-known Kuranishi method [Okonek andTeleman 1995a; 1996a; Donaldson and Kronheimer 1990; Lubke and Teleman1995]. More precisely: The linearization of the PU(2)-monopole equations in asolution p = (A,Ψ) defines an elliptic deformation complex

0→ A0(|g0(P u))D0p→ A1(|g0(P u))⊕A0(Σ+(P u))

D1p→ A2

+(|g0(P u))⊕A0(Σ−(P u))→ 0

whose differentials are given by D0p(f) = (−dAf, fΨ) and

D1p(α, ψ) = (d+

Aα− Γ−1[m(ψ,Ψ) +m(Ψ, ψ)], 6DA,aψ + γ(α)Ψ).


Here m denotes the sesquilinear map associated with the quadratic map µ00. LetHip, for i = 0, 1, 2, denote the harmonic spaces of the elliptic complex above. Thestabilizer G0p of the point p ∈ A(δ(P u)) × A0(Σ+(P u)) is a finite dimensionalLie group, isomorphic to a closed subgroup of SU(2), which acts in a naturalway on the spaces Hip.

Proposition 2.2.1 [Okonek and Teleman 1996a; Teleman 1997]. For everypoint p ∈Mσ

a there exists a neighborhood Vp ⊂Mσa , a G0p-invariant neighborhood

Up of 0 ∈ H1p, an G0p-equivariant map Kp : Up −→ H2

p with Kp(0) = 0 anddKp(0) = 0, and an isomorphism of ringed spaces

Vp ' Z(Kp)/G0p

sending p to [0].

The local isomorphisms Vp ' Z(Kp)/G0p define the structure of a smooth man-

ifold on the open subset

Mσa,reg := [A,Ψ] ∈Mσ

a : G0p = 1, H2p = 0,

and a real analytic orbifold structure in the open set of points p ∈ Mσa with

G0p finite. The dimension of Mσa,reg coincides with the expected dimension of

the PU(2)-monopole moduli space, which is given by the index χ(SWσa) of the

elliptic deformation complex:

χ(SWσa ) = 1

2 (−3p1(δ(P u)) + c1(detP u)2)− 12(3e(X) + 4σ(X)).

Our next goal is to describe the fixed point set of the S1-action on Mσa intro-

duced above.First consider the closed subspace D(δ(P u)) ⊂Mσ

a of points of the form [A, 0].It can be identified with the Donaldson moduli space of anti-selfdual connectionsin the PU(2)-bundle δ(P u) modulo the gauge group G0. Note however, that ifH1(X,Z2) 6= 0, D(δ(P u)) does not coincide with the usual moduli space ofPU(2)-instantons in δ(P u) but is a finite cover of it.

The stabilizer G0p of a Donaldson point (A, 0) contains always ± id, henceMσa has at least Z2-orbifold singularities in the points of D(δ(P u)).Secondly, consider S1 as a subgroup of PU(2) via the standard embedding

S1 3 ζ 7−→[(ζ0

01

)]∈ PU(2). Note that any S1-reduction ρ : P −→ δ(P u)

of δ(P u) defines a reduction τρ : P ρ := P u ×δ(Pu) P −→ P uσ−→ Pg of the

SpinU(2)-structure σ to a SpinS1×S1

-structure, hence a pair of Spinc-structuresτ iρ : P ρi −→ Pg. One has natural isomorphisms

detP ρ1 ⊗ detP ρ2 = (detP u)⊗2 , detP ρ1 ⊗ (detP ρ2)−1 = P⊗2 ,

and natural embeddings Σ±(P ρi) −→ Σ±(P u) induced by the bundle morphismP ρi −→ P u. A pair (A,Ψ) will be called abelian if it lies in the image ofA(P )×A0(Σ+(P ρ1)) for a suitable S1-reduction ρ of δ(P u).


Proposition 2.2.2. The fixed point set of the S1-action on Mσa is the union of

the Donaldson locus D(δ(P u)) and the locus of abelian solutions. The latter canbe identified with the disjoint union∐

ρ

Wτ1ρi

2πFa,

where the union is over all S1-reductions of the PU(2)-bundle δ(P u).

This result suggests to use the S1-quotient of Mσa \ (Mσ

a)S1

for the comparisonof Donaldson invariants and (twisted) Seiberg–Witten invariants, as explainedin [Okonek and Teleman 1996a].

Note that only using moduli spaces Mσθ of quaternionic monopoles one gets,

by the proposition above, moduli spaces of non-twisted abelian monopoles in thefixed point locus of the S1-action. This was one of the motivations for studyingthe quaternionic monopole equations in [Okonek and Teleman 1996a]. There ithas been shown that one can use the moduli spaces of quaternionic monopolesto relate certain Spinc-polynomials to the original nontwisted Seiberg–Witteninvariants.

The remainder of this section is devoted to the description of the Uhlenbeckcompactification of the moduli spaces of PU(2)-monopoles [Teleman 1996].

First of all, the Weitzenbock formula and the maximum principle yield abound on the spinor component, as in the abelian case. More precisely, one hasthe a priori estimate

supX|Ψ|2 ≤ Cg,a := max

(0, C sup(−1

2s+ |F+

a |))

on the space of solutions of (SWσa ), where C is a universal positive constant.

The construction of the Uhlenbeck compactification of Mσa is based, as in the

instanton case, on the following three essential results.

1. A compactness theorem for the subspace of solutions with suitable bounds onthe curvature of the connection component.

2. A removable singularities theorem.3. Controlling bubbling phenomena for an arbitrary sequence of points in the

moduli space Mσa .

1. A compactness result.

Theorem 2.2.3. There exists a positive number δ > 0 such that for everyoriented Riemannian manifold (Ω, g) endowed with a SpinU(2)(4)-structure σ :P u −→ Pg and a fixed connection a ∈ A(detP u), the following holds:

If (An,Ψn) is a sequence of solutions of (SWσa), such that any point x ∈ Ω

has a geodesic ball neighborhood Dx with∫Dx

|FAn |2 < δ2


for all large enough n, then there is a subsequence (nm) ⊂ N and gauge transfor-mations fm ∈ G0 such that f∗m (Anm,Ψnm) converges in the C∞-topology on Ω.

2. Removable singularities. Let g be a metric on the 4-ball B, and let

σ : P u = B × SpinU(2)(4) −→ Pg ' B × SO(4)

be a SpinU(2)-structure in (B, g). Fix a ∈ iA1B and put B• := B\0, σ• := σ|B• .

Theorem 2.2.4. Let (A0,Ψ0) be a solution of the equations (SWσ•

a ) on thepunctured ball such that

‖FA0‖2L2 <∞.

Then there exists a solution (A,Ψ) of (SWσa) on B and a gauge transformation

f ∈ C∞(B•, SU(2)) such that f∗(A|B• ,Ψ|B•) = (A0,Ψ0).

3. Controlling bubbling phenomena. The main point is that the selfdual com-ponents F+

Anof the curvatures of a sequence of solutions ([An,Ψn])n∈N in Mσ

a

cannot bubble.

Definition 2.2.5. Let σ : P u −→ Pg be a SpinU(2)-structure in (X, g) and fixa ∈ A(detP u). An ideal monopole of type (σ, a) is a pair ([A′,Ψ′], x1, . . . , xl)consisting of a point [A′,Ψ′] ∈M

σ′la , where σ′l : P ′u −→ Pg is a SpinU(2)-structure

satisfying

detP ′u = detP u , p1(δ(P ′u)) = p1(δ(P u)) + 4l,

and x1, . . . , xl ∈ SlX. The set of ideal monopoles of type (σ, a) is

IMσa :=

∐l≥0

Mσ′la × SlX.

Theorem 2.2.6. There exists a metric topology on IMσa such that the moduli

space Mσa becomes an open subspace with compact closure Mσ

a .

Sketch of proof. Given a sequence ([An,Ψn])n∈N of points in Mσa , one finds

a subsequence ([Anm,Ψnm])m∈N , a finite set of points S ⊂ X, and gauge trans-formations fm such that (Bm,Φm) := f∗m(Anm ,Ψnm) converges on X \ S in theC∞-topology to a solution (A0,Ψ0). This follows from the compactness theoremabove, using the fact that the total volume of the sequence of measures |FAn|2is bounded. The set S consists of points in which the measure |FAnm |2 becomesconcentrated as m tends to infinity.

By the Removable Singularities theorem, the solution (A0,Ψ0) extends aftergauge transformation to a solution (A,Ψ) of (SWσ′

a ) on X, for a possibly differentSpinU(2)-structure σ′ with the same determinant line bundle. The curvature ofA satisfies

|FA|2 = limm→∞

|FAnm |2 − 8π2

∑x∈S

λxδx,


where δx is the Dirac measure of the point x. Now it remains to show that theλx’s are natural numbers and∑

x∈Sλx = 1

4(p1(δ(P ′u))− p1(δ(P u))).

This follows as in the instanton case, if one uses the fact that the measures|F+Anm|2 cannot bubble in the points x ∈ S as m → ∞ and that the integral

of |F−An |2 − |F+

An|2 is a topological invariant of δ(P u). In this way one gets an

ideal monopole m := ([A,Ψ], λ1x1, . . . , λkxk) of type (σ, a). With respect to asuitable topology on the space of ideal monopoles, one has lim

m→∞[Anm,Ψnm] = m.

3. Seiberg–Witten Theory and Kahler Geometry

3.1. Monopoles on Kahler Surfaces. Let (X, J, g) be an almost Hermitiansurface with associated Kahler form ωg. We denote by Λpq the bundle of (p, q)-forms on X and by Apq its space of sections. The Hermitian structure definesan orthogonal decomposition

Λ2+ ⊗C = Λ20 ⊕ Λ02 ⊕ Λ00ωg

and a canonical Spinc-structure τ . The spinor bundles of τ are

Σ+ = Λ00 ⊕ Λ02 , Σ− = Λ01 ,

and the Chern class of τ is the first Chern class c1(T 10J ) = c1(K∨X) of the complex

tangent bundle. The complexification of the canonical Clifford map γ is thestandard isomorphism

γ : Λ1 ⊗ C −→ Hom(Λ00 ⊕Λ02,Λ01), γ(u)(ϕ + α) =√

2(ϕu01 − iΛgu10 ∧ α),

and the induced isomorphism Γ : Λ20 ⊕ Λ02 ⊕ Λ00ωg −→ End0(Λ00 ⊕ Λ02) actsby

(λ20, λ02, fωg) Γ7−→ 2[−if − ∗ (λ20 ∧ ·)λ02 ∧ · if

]∈ End0(Λ00 ⊕Λ02).

Recall from Section 1.1 that the set Spinc(X) of equivalence classes of Spinc-structures in (X, g) is a H2(X,Z)-torsor. Using the class of the canonical Spinc-structure c := [τ ] as base point, Spinc(X) can be identified with the set ofisomorphism classes of S1-bundles: When M is an S1-bundle with c1(M) = m,the Spinc-structure τm has spinor bundles Σ± ⊗M and Chern class 2c1(M) −c1(KX). Let cm be the class of τm.

Suppose now that (X, J, g) is Kahler, and let k ∈ A(KX) be the Chern connec-tion in the canonical line bundle. In order to write the (abelian) Seiberg–Wittenequations associated with the Spinc-structure τm in a convenient form, we makethe variable substitution a = k⊗e⊗2 for a connection e ∈ A(M) in the S1-bundleM , and we write the spinor Ψ as a sum Ψ = ϕ + α ∈ A0(M)⊕A02(M).


Lemma 3.1.1 [Witten 1994; Okonek and Teleman 1996b]. Let (X, g) be a Kahlersurface, β ∈ A11

R a closed real (1,1)-form in the de Rham cohomology class b,and let M be a S1-bundle with (2c1(M) − c1(KX) − b)·[ωg] < 0. The pair(k⊗ e⊗2, ϕ+α) ∈ A(det(Σ+⊗M))×A0(Σ+⊗M) solves the equations (SWτm

β )if and only if α = 0, F 20

e = F 02e = 0, ∂eϕ = 0, and

iΛgFe + 14ϕϕ + (1

2s− πΛgβ) = 0. (∗)

Note that the conditions F 20e = F 02

e = 0, ∂eϕ = 0 mean that e is the Chernconnection of a holomorphic structure in the Hermitian line bundleM and that ϕis a holomorphic section with respect to this holomorphic structure. Integratingthe relation (∗) and using the inequality in the hypothesis, one sees that ϕ cannotvanish identically.

To interpret the condition (∗) consider an arbitrary real valued function func-tion t : X −→ R, and let

mt : A(M)×A0(M) −→ iA0

be the map defined by

mt(e, ϕ) := ΛgFe − 14iϕϕ + it.

It is easy to see that (after suitable Sobolev completions) A(M)×A0(M) hasa natural symplectic structure, and that mt is a moment map for the action ofthe gauge group G = C∞(X, S1). Let GC = C∞(X,C∗) be the complexificationof G, and let H ⊂ A(M)× A0(M) be the closed set

H := (e, ϕ) ∈ A(M)× A0(M) : F 02e = 0, ∂eϕ = 0

of integrable pairs. For any function t put

Ht := (e, ϕ) ∈ H : GC(e, ϕ) ∩m−1t (0) 6= ∅.

Using a general principle in the theory of symplectic quotients, which also holdsin our infinite dimensional framework, one can prove that the GC-orbit of a point(e, ϕ) ∈ Ht intersects the zero set m−1

t (0) of the moment map mt precisely alonga G-orbit (see Figure 3). In other words, there is a natural bijection of quotients

[m−1t (0) ∩H]

/G ' Ht

/GC . (1)

Now take t := −(12s − πΛgβ) and suppose again that the assumptions in

Lemma 3.1.1 hold. We have seen that m−1t (0) ∩H cannot contain pairs of the

form (e, 0), hence G (GC) acts freely on m−1t (0) ∩H (Ht). Using this fact one

can show that Ht is open in the space H of integrable pairs, and endowingthe two quotients in (1) with the natural real analytic structures, one provesthat (1) is a real analytic isomorphism. By the lemma, the first quotient isprecisely the moduli space Wτm

β . The second quotient is a complex-geometricobject, namely an open subspace in the moduli space of simple holomorphicpairs H ∩ ϕ 6= 0

/GC . A point in this moduli space can be regarded as an


GC (e, ϕ)

m−1t (0)

G(e′, ϕ′)

Figure 3.

isomorphism class of pairs (M, ϕ) consisting of a holomorphic line bundle M oftopological type M , and a holomorphic section in M. Such a pair defines a pointin Ht

/GC if and only if M admits a Hermitian metric h satisfying the equation

iΛFh + 14ϕϕ

h = t.

This equation for the unknown metric h is the vortex equation associated withthe function t: it is solvable if and only if the stability condition

(2c1(M) − c1(KX)− b)·[ωg] < 0

is fulfilled. Let Dou(m) be the Douady space of effective divisors D ⊂ X withc1(OX(D)) = m. The map Z : Ht

/GC −→ Dou(m) which associates to an orbit

[e, ϕ] the zero-locus Z(ϕ) ⊂ X of the holomorphic section ϕ is an isomorphismof complex spaces.

Putting everything together, we have the following interpretation for themonopole moduli spaces Wτm

β on Kahler surfaces.

Theorem 3.1.2 [Okonek and Teleman 1995a; 1996b]. Let (X, g) be a compactKahler surface, and let τm be the Spinc-structure defined by the S1-bundle M .Let β ∈ A11

R be a closed 2-form representing the de Rham cohomology class bsuch that

(2c1(M) − c1(KX)− b)·[ωg] < 0 (alternatively , > 0).


If c1(M) 6∈ NS(X), were NS(X) is the Neron–Severi group of X, then Wτmβ = ∅.

When c1(M) ∈ NS(X), there is a natural real analytic isomorphism

Wτmβ ' Dou(m) (respectively , Dou(c1(KX) −m)).

A moduli space Wτmβ 6= ∅ is smooth at the point corresponding to D ∈ Dou(m)

if and only if h0(OD(D)) = dimD Dou(m). This condition is always satisfiedwhen b1(X) = 0. If Wτm

β is smooth at a point corresponding to D ∈ Dou(m),then it has the expected dimension in this point if and only if h1(OD(D)) = 0.

The natural isomorphisms Wτmβ ' Dou(m) respects the orientations induced

by the complex structure ofX when (2c1(M)−c1(KX)−b)·[ωg] < 0. If (2c1(M)−c1(KX)−b)·[ωg] > 0, then the isomorphism Wτm

β ' Dou(c1(KX)−m) multipliesthe complex orientations by (−1)χ(M) [Okonek and Teleman 1996b].

Example. Consider again the complex projective plane P2, polarized by

h = c1(OP2(1)).

The expected dimension of Wτmβ is m(m + 3h). The theorem above yields the

following explicit description of the corresponding moduli spaces:

Wτmβ '

|OP2(m)| if (2m+ 3h− [β])·h < 0,|OP2(−(m+ 3))| if (2m+ 3h− [β])·h > 0.

E. Witten [1994] has shown that on Kahlerian surfaces X with geometric genuspg > 0 all nontrivial Seiberg–Witten invariants SWX,O(c) satisfy wc = 0.

In the case of Kahlerian surfaces with pg = 0 one has a different situation.Suppose for instance that b1(X) = 0. Choose the standard orientation O1 ofH1(X,R) = 0 and the component H0 containing Kahler classes to orient themoduli spaces of monopoles. Then, using the previous theorem and the wall-crossing formula, we get:

Proposition 3.1.3. Let X be a Kahler surface with pg = 0 and b1 = 0. Ifm ∈ H2(X,Z) satisfies m(m − c1(KX)) ≥ 0, that is, the expected dimensionw2m−c1(KX) is nonnegative, then

SW+X,H0

(cm) =

1 if Dou(m) 6= ∅,0 if Dou(m) = ∅,

SW−X,H0(cm) =

0 if Dou(m) 6= ∅,−1 if Dou(m) = ∅.

Our next goal is to show that the PU(2)-monopole equations on a Kahler sur-face can be analyzed in a similar way. This analysis yields a complex geometricdescription of the moduli spaces whose S1-quotients give formulas relating theDonaldson invariants to the Seiberg–Witten invariants. If the base is projective,one also has an algebro-geometric interpretation [Okonek et al. 1999], whichleads to explicitly computable examples of moduli spaces of PU(2)-monopoles


[Teleman 1996]. Such examples are important, because they illustrate the gen-eral mechanism for proving the relation between the two theories, and help tounderstand the geometry of the ends of the moduli spaces in the more difficultC∞-category.

Recall that, since (X, g) comes with a canonical Spinc-structure τ , the data ofof a SpinU(2)-structure in (X, g) is equivalent to the data of a Hermitian bundleE of rank 2. The bundles of the corresponding SpinU(2)-structure σ : P u −→ Pgare given by δ(P u) = PE

/S1, detP u = detE⊗KX , and Σ±(P u) = Σ±⊗E⊗KX .

Suppose that detP u admits an integrable connection a ∈ A(detP u). Letk ∈ A(KX) be the Chern connection of the canonical bundle, and let λ :=a⊗ k∨ be the induced connection in L := detE. We denote by L := (L, ∂λ) theholomorphic structure defined by λ. Now identify the affine space A(δ(P u)) withthe space Aλ⊗k⊗2(E⊗KX) of connections in E⊗KX which induce λ⊗k⊗2 = a⊗kin det(E ⊗KX), and identify A0(Σ+(P u)) with

A0(E ⊗KX)⊕A0(E) = A0(E ⊗KX) ⊕A02(E ⊗KX).

Proposition 3.1.4. Fix an integrable connection a ∈ A(detE ⊗KX). A pair(A, ϕ+α) ∈ Aλ⊗k⊗2(E⊗KX)× [A0(E⊗KX)⊕A02(E⊗KX)] solves the PU(2)-monopole equations (SWσ

a) if and only if A is integrable and one of the followingconditions is satisfied :

(I) α = 0, ∂Aϕ = 0, and iΛgF 0A + 1

2(ϕϕ)0 = 0.(II) ϕ = 0, ∂Aα = 0, and iΛgF 0

A − 12 ∗ (α ∧ α)0 = 0.

Note that solutions (A, ϕ) of type I give rise to holomorphic pairs (FA, ϕ), con-sisting of a holomorphic structure in F := E ⊗K and a holomorphic section ϕ

in FA. The remaining equation iΛgF 0A + 1

2(ϕϕ)0 = 0 can again be interpretedas the vanishing condition for a moment map for the G0-action in the space ofpairs (A, ϕ) ∈ Aλ⊗k⊗2(F ) × A0(F ). We shall study the corresponding stabilitycondition in the next section.

The analysis of the solutions of type II can be reduced to the investigation ofthe type I solutions: Indeed, if ϕ = 0 and α ∈ A02(E ⊗KX) satisfies ∂Aα = 0,we see that the section ψ := α ∈ A0(E) must be holomorphic, that is, it satisfies∂A⊗[a∨]ψ = 0. On the other hand one has − ∗ (α ∧ α)0 = ∗(α ∧ ¯α)0 = (ψψ)0.

3.2. Vortex Equations and Stable Oriented Pairs. Let (X, g) be a com-pact Kahler manifold of arbitrary dimension n, and let E be a differentiable vec-tor bundle of rank r, endowed with a fixed holomorphic structure L := (L, ∂L)in L := detE.

An oriented pair of type (E,L) is a pair (E, ϕ), consisting of a holomor-phic structure E = (E, ∂E) in E with ∂det E = ∂L, and a holomorphic sectionϕ ∈ H0(E). Two oriented pairs are isomorphic if they are equivalent under thenatural action of the group SL(E) of differentiable automorphisms of E withdeterminant 1.


An oriented pair (E, ϕ) is simple if its stabilizer in SL(E) is contained in thecenter Zr · idE of SL(E); it is strongly simple if this stabilizer is trivial.

Proposition 3.2.1 [Okonek and Teleman 1996a]. There exists a (possiblynon-Hausdorff ) complex analytic orbifold Msi(E,L) parameterizing isomorphismclasses of simple oriented pairs of type (E,L). The open subset Mssi(E,L) ⊂Msi(E,L) of classes of strongly simple pairs is a complex analytic space, and thepoints Msi(E,L) \Mssi(E,L) have neighborhoods modeled on Zr-quotients.

Now fix a Hermitian background metricH inE. In this section we use the symbol(SU(E)) U(E) for the groups of (special) unitary automorphisms of (E,H), andnot for the bundles of (special) unitary automorphisms.

Let λ be the Chern connection associated with the Hermitian holomorphicbundle (L, detH). We denote by A∂λ(E) the affine space of semiconnectionsin E which induce the semiconnection ∂λ = ∂L in L = detE, and we writeAλ(E) for the space of unitary connections in (E,H) which induce λ in L.The map A 7−→ ∂A yields an identification Aλ(E) −→ A∂λ

(E), which endowsthe affine space Aλ(E) with a complex structure. Using this identification andthe Hermitian metric H, the product Aλ(E) × A0(E) becomes — after suitableSobolev completions — an infinite dimensional Kahler manifold. The map

m : Aλ(E)× A0(E) −→ A0(su(E))

defined by m(A, ϕ) := ΛgF 0A − i

2(ϕϕ)0 is a moment map for the SU(E)-actionon the Kahler manifold Aλ(E)× A0(E).

We denote by Hλ(E) := (A, ϕ) ∈ Aλ(E) × A0(E)| F 02A = 0, ∂Aϕ = 0 the

space of integrable pairs, and by Hλ(E)si the open subspace of pairs

(A, ϕ) ∈ Hλ(E)

with (∂A, ϕ) simple. The quotient

Vλ(E) :=(Hλ(E) ∩m−1(0)

)/SU(E)

is called the moduli space of projective vortices, and

V∗λ(E) :=(Hsiλ (E) ∩m−1(0)

)/SU(E)

is called the moduli space of irreducible projective vortices. Note that a vortex(A, ϕ) is irreducible if and only if SL(E)(A,ϕ) ⊂ Zr idE . Using again an infinitedimensional version of the theory of symplectic quotients (as in the abelian case),one gets a homeomorphism

j : Vλ(E) '−→ Hpsλ (E)

/SL(E)

where Hpsλ (E) is the subspace of Hλ(E) consisting of pairs whose SL(E)-orbit

meets the vanishing locus of the moment map. Hpsλ (E) is in general not open,


but Hsλ(E) := H

psλ (E) ∩ Hsi

λ (E) is open, and restricting j to V∗λ(E) yields anisomorphism of real analytic orbifolds

V∗λ(E) '−→ Hsλ(E)

/SL(E) ⊂Msi(E,L).

The imageMs(E,L) := Hs

λ(E)/

SL(E)

of this isomorphism can be identified with the set of isomorphism classes ofsimple oriented holomorphic pairs (E, ϕ) of type (E,L), with the property thatE admits a Hermitian metric with deth = detH which solves the projectivevortex equation

iΛgF 0h + 1

2(ϕϕh)0 = 0.

Here Fh is the curvature of the Chern connection of (E, h).The set Ms(E,L) has a purely holomorphic description as the subspace of

elements [E, ϕ] ∈Msi(E,L) which satisfy a suitable stability condition.This condition is rather complicated for bundles E of rank r > 2, but it

becomes very simple when r = 2.Recall that, for any torsion free coherent sheaf F 6= 0 over a n-dimensional

Kahler manifold (X, g), one defines the g-slope of F by

µg(F) :=c1(det F) ∪ [ωg]n−1

rk(F).

A holomorphic bundle E over (X, g) is called slope-stable if µg(F) < µg(E)for all proper coherent subsheaves F ⊂ E. The bundle E is slope-polystable if itdecomposes as a direct sum E = ⊕Ei of slope-stable bundles with µg(Ei) = µg(E).

Definition 3.2.2. Let (E, ϕ) be an oriented pair of type (E,L) with rkE = 2over a Kahler manifold (X, g). The pair (E, ϕ) is stable if ϕ = 0 and E is slope-stable, or ϕ 6= 0 and the divisorial component Dϕ of the zero-locus Z(ϕ) ⊂ X

satisfies µg(OX(Dϕ)) < µg(E). The pair (E, ϕ) is polystable if it is stable orϕ = 0 and E is slope-polystable.

Example. Let D ⊂ X be an effective divisor defined by a section

ϕ ∈ H0(OX(D)) \ 0,

and put E := OX(D) ⊕ [L ⊗ OX(−D)]. The pair (E, ϕ) is stable if and only ifµg(OX(2D)) < µg(L).

The following result gives a metric characterization of polystable oriented pairs.

Theorem 3.2.3 [Okonek and Teleman 1996a]. Let E be a differentiable vectorbundle of rank 2 over (X, g) endowed with a Hermitian holomorphic structure(L, l) in detE. An oriented pair of type (E,L) is polystable if and only if E

admits a Hermitian metric h with det h = l which solves the projective vortexequation

iΛgF 0h + 1

2(ϕϕh)0 = 0.


If (E, ϕ) is stable, then the metric h is unique.

This result identifies the subspace Ms(E,L) ⊂ Msi(E,L) as the subspace ofisomorphism classes of stable oriented pairs.

Theorem 3.2.3 can be used to show that the moduli spaces Mσa of PU(2)-

monopoles on a Kahler surface have a natural complex geometric descriptionwhen the connection a is integrable. Recall from Section 3.1 that in this caseMσa decomposes as the union of two Zariski-closed subspaces

Mσa = (Mσ

a )I ∪ (Mσa)II

according to the two conditions I, II in Proposition 3.1.4. By this proposi-tion, both terms of this union can be identified with moduli spaces of projectivevortices. Using again the symbol ∗ to denote subsets of points with central sta-bilizers, one gets the following Kobayashi–Hitchin type description of (Mσ

a )∗ interms of stable oriented pairs.

Theorem 3.2.4 [Okonek and Teleman 1996a; 1997]. If a ∈ A(detP u) is inte-grable, the moduli space Mσ

a decomposes as a union Mσa = (Mσ

a)I ∪ (Mσa)II of

two Zariski closed subspaces isomorphic with moduli spaces of projective vortices,which intersect along the Donaldson moduli space D(δ(P u)). There are naturalreal analytic isomorphisms

(Mσa )∗I

'−→ Ms(E ⊗KX ,L⊗K⊗2X ), (Mσ

a )∗II'−→ Ms(E∨,L∨),

where L denotes the holomorphic structure in detE = detP u ⊗K∨X defined by∂a and the canonical holomorphic structure in KX .

Example (R. Plantiko). On P2, endowed with the standard Fubini–Study met-ric g, we consider the SpinU(2)(4)-structure σ : P u −→ Pg defined by the stan-dard Spinc(4)-structure τ : P c −→ Pg and the U(2)-bundle E with c1(E) = 7,c2(E) = 13, and we fix an integrable connection a ∈ A(detP u). This SpinU(2)(4)-structure is characterized by c1(det(P u)) = 4, p1(δ(P u)) = −3, and the bundleF := E ⊗KP2 has Chern classes c1(F ) = 1, c2(F ) = 1. It is easy to see thatevery stable oriented pair (F, ϕ) of type (F,OP2(1)) with ϕ 6= 0 fits into an exactsequence of the form

0 −→ Oϕ−→ F −→ JZ(ϕ) ⊗ OP2(1) −→ 0,

where F = TP2(−1) and the zero locus Z(ϕ) of ϕ consists of a simple pointzϕ ∈ P2. Two such pairs (F, ϕ), (F, ϕ′) define the same point in the modulispace Ms(F,OP2(1)) if and only if ϕ′ = ±ϕ. The resulting identification

Ms(F,OP2(1)) = H0(TP2(−1))/± id

is a complex analytic isomorphism.


Since every polystable pair of type (F,OP2(1)) is actually stable, and sincethere are no polystable oriented pairs of type (E∨,OP2(−7)), Theorem 3.2.4yields a real analytic isomorphism

Mσa = H0(TP2(−1))

/± id,

where the origin corresponds to the unique stable oriented pair of the form(TP2(−1), 0). The quotient H0(TP2(−1))

/± id has a natural algebraic com-

pactification C, given by the cone over the image of P(H0(TP2(−1))) under theVeronese map to P(S2H0(TP2(−1))). This compactification coincides with theUhlenbeck compactification Mσ

a (see Section 2.2 and [Teleman 1996]). More pre-cisely, let σ′ : P ′u −→ Pg be the SpinU(2)(4)-structure with detP ′u = detP u

and p1(P ′u) = 1. This structure is associated with τ and the U(2)-bundle E′

with Chern classes c1(E′) = 7, c2(E′) = 12. The moduli space Mσ′

a consistsof one (abelian) point, the class of the abelian solution corresponding to thestable oriented pair (OP2 ⊕ OP2(1), idOP2

) of type (E′ ⊗KP2 ,OP2(1)). Mσ′

a canbe identified with the moduli space

Wτi

2πFa

of [ i2πFa]-twisted abelian Seiberg–Witten monopoles. Under the identificationC = Mσ

a , the vertex of the cone corresponds to the unique Donaldson pointwhich is given by the stable oriented pair (TP2(−1), 0). The base of the conecorresponds to the space Mσ′

a ×P2 of ideal monopoles concentrated in one point.

We close this section by explaining the stability concept which describes thesubset Ms

X(E,L) ⊂MsiX(E,L) in the general case r ≥ 2. This stability concept

does not depend on the choice of parameter and the corresponding moduli spacescan be interpreted as “master spaces” for holomorphic pairs (see next section);in the projective framework they admit Gieseker type compactifications [Okoneket al. 1999].

We shall find this stability concept by relating the SU(E)-moment map

m : Aλ(E)× A0(E) −→ A0(su(E))

to the universal family of U(E)-moment maps mt : A(E)×A0(E) −→ A0(u(E))defined by

mt(A, ϕ) := ΛgFA − 12i(ϕϕ) + 1

2it idE ,

where t ∈ A0 is an arbitrary real valued function. Given t, we consider thesystem of equations

F 02A = 0,

∂Aϕ = 0,

iΛgFA + 12(ϕϕ) = 1

2 t idE

(Vt)


for pairs (A, ϕ) ∈ A(E)× A0(E). Put

ρt :=1

4πn

∫X

tωng .

To explain our first result, we have to recall some classical stability concepts forholomorphic pairs.

For any holomorphic bundle E over (X, g) denote by S(E) the set of reflexivesubsheaves F ⊂ E with 0 < rk(F) < rk(E), and for a fixed section ϕ ∈ H0(E) put

Sϕ(E) := F ∈ S(E) : ϕ ∈ H0(F).

Define real numbers mg(E) and mg(E, ϕ) by

mg(E) := max(µg(E), supF′∈S(E)

µg(F′)), mg(E, ϕ) := infF∈Sϕ(E)

µg(E/F).

A bundle E is ϕ-stable in the sense of S. Bradlow when mg(E) < mg(E, ϕ). Letρ ∈ R be any real parameter. A holomorphic pair (E, ϕ) is called ρ-stable if ρsatisfies the inequality

mg(E) < ρ < mg(E, ϕ).

The pair (E, ϕ) is ρ-polystable if it is ρ-stable or E-splits holomorphically asE = E′⊕E′′ such that ϕ ∈ H0(E′), (E′, ϕ) is ρ-stable and E′′ is a slope-polystablevector bundle with µg(E′′) = ρ [Bradlow 1991]. Let GL(E) be the group ofbundle automorphisms of E. With these definitions one proves [Okonek andTeleman 1995a] (see [Bradlow 1991] for the case of a constant function t):

Proposition 3.2.5. The complex orbit GL(E)·(A, ϕ) of an integrable pair(A, ϕ) ∈ A(E)×A0(E) contains a solution of (Vt) if and only if the pair (EA, ϕ)is ρt-polystable.

Now fix again a Hermitian metric H in E and an integrable connection λ in theHermitian line bundle L := (detE, detH). Consider the system of equations

F 02A = 0,

∂Aϕ = 0,

iΛgF 0A + 1

2(ϕϕ)0 = 0

(V 0)

for pairs (A, ϕ) ∈ Aλ(E)× A0(E). Then one can prove

Proposition 3.2.6. Let (A, ϕ) ∈ Aλ(E) × A0(E) be an integrable pair . Thefollowing assertions are equivalent :

(i) The complex orbit SL(E)·(A, ϕ) contains a solution of (V 0).(ii) There exists a function t ∈ A0 such that the GL(E)-orbit GL(E)·(A, ϕ)

contains a solution of (Vt).(iii) There exists a real number ρ such that the pair (EA, ϕ) is ρ-polystable.


Corollary 3.2.7. The open subspace Ms(E,L) ⊂ Msi(E,L) is the set ofisomorphism classes of simple oriented pairs which are ρ-polystable for someρ ∈ R.

Remark 3.2.8. There exist stable oriented pairs (E, ϕ) whose stabilizer withrespect to the GL(E)-action is of positive dimension. Such pairs cannot beρ-stable for any ρ ∈ R.

Note that the moduli spaces Ms(E,L) have a natural C∗-action defined byz ·[E, ϕ] := [E, z1/rϕ]. This action is well defined since r-th roots of unity arecontained in the complex gauge group SL(E).

There exists an equivalent definition for stability of oriented pairs, which doesnot use the parameter dependent stability concepts of [Bradlow 1991]. The factthat it is expressible in terms of ρ-stability is related to the fact that the modulispaces Ms(E,L) are master spaces for moduli spaces of ρ-stable pairs.

3.3. Master Spaces and the Coupling Principle. Let X ⊂ PNC be asmooth complex projective variety with hyperplane bundle OX(1). All degreesand Hilbert polynomials of coherent sheaves will be computed corresponding tothese data.

We fix a torsion-free sheaf E0 and a holomorphic line bundle L0 over X, andwe choose a Hilbert polynomial P0. By PF we denote the Hilbert polynomialof a coherent sheaf F. Recall that any nontrivial torsion free coherent sheaf F

admits a unique subsheaf Fmax for which PF′/rk F′ is maximal and whose rankis maximal among all subsheaves F′ with PF′/rk F′ maximal.

An L0-oriented pair of type (P0,E0) is a triple (E, ε, ϕ) consisting of a torsionfree coherent sheaf E with determinant isomorphic to L0 and Hilbert polynomialPE = P0, a homomorphism ε : det E −→ L0, and a morphism ϕ : E −→ E0.The homomorphisms ε and ϕ will be called the orientation and the framing ofthe oriented pair. There is an obvious equivalence relation for such pairs. Whenkerϕ 6= 0, we set

δE,ϕ := PE −rk E

rk[ker(ϕ)max

]Pker(ϕ)max .

An oriented pair (E, ε, ϕ) is semistable if either

1. ϕ is injective, or2. ε is an isomorphism, kerϕ 6= 0, δE,ϕ ≥ 0, and for all nontrivial subsheaves

F ⊂ E the inequality

PF

rk F− δE,ϕ

rk F≤ PE

rk E− δE,ϕ

rk E.

holds.

The corresponding stability concept is slightly more complicated; see [Okoneket al. 1999]. Note that the (semi)stability definition above does not depend on


a parameter. It is, however, possible to express (semi)stability in terms of theparameter dependent Gieseker-type stability concepts of [Huybrechts and Lehn1995]. For example, (E, ε, ϕ) is semistable if and only if ϕ is injective, or E isGieseker semistable, or there exists a rational polynomial δ of degree smallerthan dimX with positive leading coefficient, such that (E, ϕ) is δ-semistable inthe sense of [Huybrechts and Lehn 1995].

For all stability concepts introduced so far there exist analogous notions ofslope-(semi)stability. In the special case when the reference sheaf E0 is the trivialsheaf OX , slope stability is the algebro-geometric analog of the stability conceptassociated with the projective vortex equation.

Theorem 3.3.1 [Okonek et al. 1999]. There exists a projective scheme

Mss(P0,E0,L0)

whose closed points correspond to gr-equivalence classes of Gieseker semistableL0-oriented pairs of type (P0,E0). This scheme contains an open subschemeMs(P0,E0,L0) which is a coarse moduli space for stable L0-oriented pairs.

It is also possible to construct moduli spaces for stable oriented pairs where theorienting line bundle is allowed to vary [Okonek et al. 1999]. This generalizationis important in connection with Gromov–Witten invariants for Grassmannians[Bertram et al. 1996].

Note that Mss(P0,E0,L0) possesses a natural C∗-action, given by

z ·[E, ε, ϕ] := [E, ε, zϕ],

whose fixed point set can be explicitly described. The fixed point locus

[Mss(P0,E0,L0)]C∗

contains two distinguished subspaces, M0 defined by the equation ϕ = 0, andM∞ defined by ε = 0. M0 can be identified with the Gieseker scheme Mss(P,L0)of equivalence classes of semistable L0-oriented torsion free coherent sheaveswith Hilbert polynomial P0. The subspace M∞ is the Grothendieck Quot-scheme QuotE0,L0

PE0−P0of quotients of E0 with fixed determinant isomorphic with

(det E0) ⊗L∨0 and Hilbert polynomial PE0 − P0.In the terminology of [Bia lynicki-Birula and Sommese 1983], M0 is the source

Msource of the C∗-space Mss(P0,E0,L0), and M∞ is its sink when nonempty.The remaining subspace of the fixed point locus

MR := [Mss(P0,E0,L0)]C∗\ [M0 ∪M∞],

the so-called space of reductions, consists of objects which are of the same typebut essentially of lower rank.

Note that the Quot scheme M∞ is empty if rk(E0) is smaller than the rank rof the sheaves E under consideration, in which case the sink of the moduli spaceis a closed subset of the space of reductions.


Recall from [Bia lynicki-Birula and Sommese 1983] that the closure of a generalC∗-orbit connects a point in Msource with a point in Msink, whereas closures ofspecial orbits connect points of other parts of the fixed point set.

The flow generated by the C∗-action can therefore be used to relate dataassociated with M0 to data associated with M∞ and MR.

The technique of computing data on M0 in terms of MR and M∞ is a verygeneral principle which we call coupling and reduction. This principle has al-ready been described in a gauge theoretic framework in Section 2.2 for relatingmonopoles and instantons. However, the essential ideas may probably be bestunderstood in an abstract Geometric Invariant Theory setting, where one has avery simple and clear picture.

Let G be a complex reductive group, and consider a linear representationρA : G −→ GL(A) in a finite dimensional vector space A. The induced actionρA : G −→ Aut(P(A)) comes with a natural linearization in OP(A)(1), hence wehave a stability concept, and thus we can form the GIT quotient

M0 := P(A)ss//G.

Suppose we want to compute “correlation functions”

ΦI := 〈µI , [M0]〉;

that is, we want to evaluate suitable products of canonically defined cohomologyclasses µi on the fundamental class [M0] of M0. Usually the µi’s are slantproducts of characteristic classes of a “universal bundle” |E0 on M0 × X withhomology classes of X. Here X is a compact manifold, and |E0 comes from atautological bundle |E0 on A ×X by applying Kempf’s Descend Lemma.

The main idea is now to couple the original problem with a simpler one,and to use the C∗-action which occurs naturally in the resulting GIT quotientsto express the original correlation functions in terms of simpler data. Moreprecisely, consider another representation ρB : G −→ GL(B) with GIT quotientM∞ := P(B)ss//G. The direct sum ρ := ρA ⊕ ρB defines a naturally linearizedG-action on the projective space P(A⊕ B). We call the corresponding quotient

M := P(A ⊕B)ss//G

the master space associated with the coupling of ρA to ρB .The space M comes with a natural C∗-action, given by

z ·[a, b] := [a, z ·b],

and the union M0 ∪M∞ is a closed subspace of the fixed point locus MC∗.Now make the simplifying assumptions that M is smooth and connected,

the C∗-action is free outside MC∗ , and suppose that the cohomology classes µiextend to M. This condition is always satisfied if the µi’s were obtained bythe procedure described above, and if Kempf’s lemma applies to the pull-backbundle p∗A(|E0) and provides a bundle on M×X extending |E0.


Under these assumptions, the complement

MR := MC∗ \ (M0 ∪M∞)

is a closed submanifold of M, disjoint from M0, and M∞. We call MR themanifold of reductions of the master space. Now remove a sufficiently small S1-invariant tubular neighborhood U of MC∗ ⊂ M, and consider the S1-quotientW := [M \ U ]

/S1. This is a compact manifold whose boundary is the union

of the projectivized normal bundles P(NM0) and P(NM∞), and a differentiableprojective fiber space PR over MR. Note that in general PR has no naturalholomorphic structure. Let n0, n∞ be the complex dimensions of the fibers ofP(NM0), P(NM∞), and let u ∈ H2(W,Z) be the first Chern class of the S1-bundle dual to M \ U −→ W . Let µI be a class as above. Then, taking intoaccount orientations, we compute:

ΦI :=⟨µI , [M0]

⟩=⟨µI ∪ un0, [P(NM0)]

⟩=⟨µI ∪ un0, [P(NM∞)]

⟩−⟨µI ∪ un0, [PR]

⟩.

In this way the coupling principle reduces the calculation of the original corre-lation functions on M0 to computations on M∞ and on the manifold of reductionsMR. A particular important case occurs when the GIT problem given by ρB istrivial, that is, when P(B)ss = ∅. Under these circumstances the functions ΦIare completely determined by data associated with the manifold of reductionsMR.

Of course, in realistic situations, our simplifying assumptions are seldom sat-isfied, so that one has to modify the basic idea in a suitable way.

One of the realistic situations which we have in mind is the coupling of co-herent sheaves with morphisms into a fixed reference sheaf E0. In this case,the original problem is the classification of stable torsion-free sheaves, and thecorresponding Gieseker scheme Mss(P0,L0) of L0-oriented semistable sheaves ofHilbert polynomial P0 plays the role of the quotient M0. The correspondingmaster spaces are the moduli spaces Mss(P0,E0,L0) of semistable L0-orientedpairs of type (P0,E0).

Coupling with E0-valued homomorphisms ϕ : E −→ E0 leads to two essentiallydifferent situations, depending on the rank r of the sheaves E under consideration:

1. When rk(E0) < r, the framings ϕ : E −→ E0 can never be injective, i.e. thereare no semistable homomorphisms. This case correspond to the GIT situationM∞ = ∅.

2. As soon as rk(E0) ≥ r, the framings ϕ can become injective, and theGrothendieck schemes QuotE0,L0

PE0−P0appear in the master space Mss(P0,E0,L0).

These Quot schemes are the analoga of the quotients M∞ in the GIT situation.

In both cases the spaces of reductions are moduli spaces of objects which are ofthe same type but essentially of lower rank.


Everything can be made very explicit when the base manifold is a curveX with a trivial reference sheaf E0 = O⊕kX . In the case k < r, the masterspaces relate correlation functions of moduli spaces of semistable bundles withfixed determinant to data associated with reductions. When r = 2, k = 1, themanifold of reductions are symmetric powers of the base curve, and the couplingprinciple can be used to prove the Verlinde formula, or to compute the volumeand the characteristic numbers (in the smooth case) of the moduli spaces ofsemistable bundles.

The general case k ≥ r leads to a method for the computation of Gromov-Witten invariants for Grassmannians. These invariants can be regarded as corre-lation functions of suitable Quot schemes [Bertram et al. 1996], and the couplingprinciple relates them to data associated with reductions and moduli spaces ofsemistable bundles. In this case one needs a master space Mss(P0,E0,L) asso-ciated with a Poincare line bundle L on Pic(X) × X which set theoretically isthe union over L0 ∈ Pic(X) of the master spaces Mss(P0,E0,L0) [Okonek et al.1999]. One could try to prove the Vafa–Intriligator formula along these lines.

Note that the use of master spaces allows us to avoid the sometimes messyinvestigation of chains of flips, which occur whenever one considers the familyof all possible C∗-quotients of the master space [Thaddeus 1994; Bradlow et al.1996].

The coupling principle has been applied in two further situations.Using the coupling of vector bundles with twisted endomorphisms, A. Schmitt

has recently constructed projective moduli spaces of Hitchin pairs [Schmitt 1998].In the case of curves and twisting with the canonical bundle, his master spacesare natural compactifications of the moduli spaces introduced in [Hitchin 1987].

Last but not least, the coupling principle can also be used in certain gaugetheoretic situations:

The coupling of instantons on 4-manifolds with Dirac-harmonic spinors hasbeen described in detail in Chapter 2. In this case the instanton moduli spacesare the original moduli spaces M0, the Donaldson polynomials are the originalcorrelation functions to compute, and the moduli spaces of PU(2)-monopoles aremaster spaces for the coupling with spinors. One is again in the special situationwhere M∞ = ∅, and the manifold of reductions is a union of moduli spaces oftwisted abelian monopoles. In order to compute the contributions of the abelianmoduli spaces to the correlation functions, one has to give explicit descriptionsof the master space in an S1-invariant neighborhood of the abelian locus.

Finally consider again the Lie group G = Sp(n)·S1 and the PSp(n)-monopoleequations (SWσ

a) for a SpinSp(n)·S1(4)-structure σ : PG −→ Pg in (X, g) and an

abelian connection a in the associated S1-bundle (see Remark 2.1.2). Regardingthe compactification of the moduli space Mσ

a as master space associated withthe coupling of PSp(n)-instantons to harmonic spinors, one should get a relationbetween Donaldson PSp(n)-theory and Seiberg–Witten type theories.


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Christian Okonek

Institut fur Mathematik

Universitat Zurich

Winterthurerstraße 190

CH-8057 Zurich

Switzerland

[email protected]

Andrei Teleman

Facultatea de Matematica

Universitatea den Bucuresti

Strada Academici 14

70109 Bucarest

Rumania

[email protected]

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